On the consistency of Sobol indices with respect to stochastic ordering of model parameters

In the past decade, Sobol's variance decomposition have been used as a tool - among others - in risk management. We show some links between global sensitivity analysis and stochastic ordering theories. This gives an argument in favor of using Sobol's indices in uncertainty quantification, as one indicator among others.

uncertainty quantifiers should increase if the uncertainty on the input increases in some way.Thus, our problematic is to find for which kind of stochastic orders and under which sufficient conditions on the output function the Sobol indices behave consistently.
Roughly speaking, given two random variables X and Y , the X-Sobol index on Y is given by It is a statistical indicator of the relative impact of X on the variability of Y .If we study the impact of several independent variables X 1 , . . ., X k on Y , then the Sobol indices may be used to provide a hierarchization of the X i 's with respect to their impact on Y .It is an interesting alternative to regression coefficient, which may be difficult to interpret if the relationship between Y and the X i 's is far from linear.
In our context, the uncertainty on the input parameters is described by a random vector X = (X 1 , . . ., X k ) of R k , with the X i 's being independent.This classical hypothesis ensures that the Sobol indices are well-defined.Even if recent works deal with dependent input variables (see e.g.[16]), we restrict ourselves to the classical theory of Sobol indices.We are interested in the Sobol indices of an output function f .One of our main result is that, under some convexity or monotonicity assumptions on f , Sobol indices have a behavior which is compatible with respect to the excess wealth order or the dispersive order (see Thms. 4. 1 and 4.3).This fact confirms that Sobol indices can be used to quantify some uncertainty, even if, depending on the purpose, moment-independent approaches might be preferred to variance decomposition (see [2]).
The paper is organized as follows.In Section 2, we recall the definition of Sobol indices.In Section 3, we provide one example showing that a change of the laws of the inputs has an important impact on Sobol indices.It motivates the theoretical study on the relationship between stochastic orders and Sobol indices.Definitions and elementary properties of stochastic orders are recalled in Appendix A. In Section 4, we state our main results about the consistency of Sobol indices with respect to the dispersive order.In Section 5 we give illustration of our results.Section 6 is devoted to multivariate extensions of the results presented in Section 4. In Section 7, we give some concluding remarks.The main mathematical proofs are postponed to Appendix B.

Definition of Sobol indices
Understanding the effects of uncertainties in parameter values on a model behaviour is a crucial issue in applied sciences.Sobol indices can be used as a tool to identify the key input parameters that drives the uncertainty on the model output.Let us recall the definitions of Sobol indices.We refer to [6,25] or [12] for more details on this subject.We consider one output Y = f (X 1 , . . ., X k ) of a model given as a deterministic function f .The input parameters are uncertain and we assume that they are given as independent random variables X 1 , . . ., X k .
The function f can be expressed as its Hoeffding decomposition [26]: with The functions f α are defined inductively: for i ∈ {1, . . ., k}.Finally, let α ⊂ {1, . . ., k}; then, With these notations, we have that: The impact of X i on Y = f (X) may be measured by the Sobol index defined as for i ∈ {1, . . .k}.There are also interactions between the variables X 1 , . . ., X k , which are encoded into the functions f α , with |α| ≥ 2. The total Sobol indices take into account the impact of the interactions and are defined as for i ∈ {1, . . .k}.

A motivating example
In this section, we provide a motivating example on a classical financial risk model.Financial models are associated with a set of parameters.In a context where these parameters are not known with certainty (due to estimation error for instance), the global sensitivity analysis is useful to assess which (uncertain) input parameters mostly contribute to the uncertainty of model output, that is, which parameters have to be estimated with care.In most of our examples, we will consider truncated distributions.The use of truncated distributions is motivated by the fact that distributions of financial parameters generally have bounded supports.Definition 3.1.Let (a, b) ∈ R 2 and let X be a random variable with density function h and cumulative distribution F .The truncated distribution of X on the interval [a, b] is the conditional distribution of X given that a < X ≤ b.Its density function is then given by Table 1.Total Sobol indices of VaR (3.2) when α = 0.9.All digits are significant with a 95% probability.
N T (0.5, 2) 0.48 0.69 In what follows, we denote by N T and E T a truncated normal and a truncated exponential laws respectively.In risk management, the Value-at-risk (VaR) is a widely used risk measure of the risk of losses associated with portfolio of financial assets (such as stocks, bonds, etc.).From a mathematical point of view, if L denotes the loss associated with a portfolio of assets, then VaR α (L) is defined as the α-quantile level of this loss, i.e., where F L denotes the cumulative distribution function of L.
Let us consider a portfolio loss of the form L = S T − K where K is positive and where S T stands for the value at time T of a portfolio of financial assets.This corresponds to the loss at time T of a short position on this portfolio when the latter has been sold at time 0 for the price S 0 = K exp(−rT ) where r is a constant risk-free interest-rate.We assume that (S t ) t≥0 follows a geometric brownian motion so that its value at time T can be expressed as where W T is the value at time T of a standard Brownian motion, µ (resp.σ) is a positive drift (resp.volatility) parameter.Therefore, the α-Value-at-Risk associated with the loss L at time T is given by where S 0 ∈ R * + and where φ −1 is the normal inverse cumulative distribution function of a standard Gaussian random variable.Note that, as soon as α ≥ 0.5, the VaR expression (3.2) can be seen as a product of log-convex non-decreasing functions with respect to µ and σ.
Our interest is to quantify the sensitivity of the uncertain parameters µ and σ on the Value-at-Risk (VaR α (L)) by evaluating the total Sobol indices S Tµ and S Tσ as defined by (2.4).We then analyze the impact of a change in the input parameters law on the total Sobol indices S Tµ and S Tσ .In the numerical illustrations, we consider a VaR associated with a risk level α = 0.9 computed on a portfolio loss with the following characteristics: Table 1 compares the total Sobol indices associated with the input parameters µ and σ under different assumptions on their probability distribution.Each line of the table corresponds to a situation where the distribution of only one parameter has been modified.We notice that the distributions of µ and σ have greater variance, respectively, than the distributions of µ * and σ * .As can be seen, changing the laws of model parameters µ and σ has a significant impact on the values of total Sobol indices S Tµ and S Tσ .In addition, these empirical illustrations seem to indicate that the more "risky" or uncertain the parameter, the higher the corresponding total Sobol index.This behavior will be investigated theoretically using the theory of stochastic ordering.

Relationships between stochastic orders and Sobol indices
In this section, we explore how an increase of riskness (in the sense of stochastic ordering) in the input parameters may have an impact on the model output.In this respect, we consider two particular stochastic orders, that is, the excess wealth order (denoted by ew) and the dispersive order (denoted by disp), whose definitions and main properties are recalled in Appendix A.

Relationship with stochastic orders when there is no interaction
In this section, we assume that there is no interaction between the X i 's, that is, f (X) can be expressed in the following additive form: where g 1 , . . ., g k are real-valued functions and K ∈ R. It is straightforward to prove that, in that case, decomposition (2.1) reduces to so that, for any i = 1, . . ., k, the "individual" Sobol index defined by (2.3) coincides with the total Sobol index defined by (2.4).Let X * i denote another variable that will be compared to X i .We will assume X * i ≤ ew X i and study the impact of replacing X i by X * i on Sobol indices.We assume that X * i is independent of X −i and we denote by X * = (X 1 , . . ., X i−1 , X * i , X i+1 , . . ., X k ) the vector X where the ith component has been replaced by X * i and by the ith Sobol index associated with f (X * ).Hence, f (X * ) is the output where distribution of the ith input distribution has been modified, and the others left unchanged.
As we consider the excess wealth order, we assume that X i and X * i have finite means.This hypothesis may be relaxed by considering the dispersive order (see Rem. 4.2).Theorem 4.1.We assume that there is no interaction, i.e. (4.1) is satisfied.Let X * i be a random variable independent of X −i and assume that X * i ≤ ew X i and −∞ < * ≤ , where * and are the left-end points of the support of X * i and X i .If g i is a non-decreasing convex function, then S * i ≤ S i and S * j ≥ S j for j = i.The proof of Theorem 4.1 is postponed in Appendix B. Remark 4.2.Note that, from Proposition A.3, the previous result also holds for any non-decreasing convex or non-increasing concave function g i as soon as X * i ≤ disp X i and X * i ≤ st X i .In addition, it is shown in [23] that if X * i ≤ disp X i with common and finite left end points of their support (i.e., * = ) then X * i ≤ st X i .

Relation with stochastic orders in presence of interactions
In the case where there are interactions, we have to consider the total Sobol indices as defined by (2.4).We will first show that the ith total Sobol indices are ordered if X * i ≤ disp X i and X * i ≤ st X i , provided that the function f is a product of functions of one variable whose log is non-decreasing and convex; indeed, we shall prove that for any α ⊂ {1, . . ., k} with i ∈ α, if X * i ≤ disp X i and X * i ≤ st X i then, α-Sobol indices are ordered.Then we consider some extensions of that case.Recall that, for α ⊂ {1, . . ., k}, the α-Sobol index is defined as: Theorem 4.3.We assume that f writes: where K ∈ R and g j , j = 1, . . ., k are real-valued functions.Let X * i be a random variable independent of X −i and assume that X * i ≤ disp X i and X * i ≤ st X i .If log g i is a non-decreasing convex or a non-increasing concave function, then (1) for any α ⊂ {1, . . ., k} with i ∈ α, we have S * α ≤ S α and for any β ⊂ {1, . . ., k} with i ∈ β, we have The proof of Theorem 4.3 is postponed in Appendix B. The conditions on stochastic ordering between X i and X * i , and on the log-convexity are necessary, as can be seen with the two counter-examples below.
Example 4.4.To see the necessity of the stochastic ordering, one can consider: 1 and it can be easily checked that S * T1 > S T1 (S * T1 ≈ 0.90 and S T1 ≈ 0.65).
Example 4.5.The log-convexity of g i is also necessary.Indeed, take: where In that case, we have S * T1 > S T1 (S * T1 ≈ 0.99 and S T1 ≈ 0.97).Now, we turn to the case where f writes as a sum of product of convex non-decreasing functions of one variable, that is, there are a finite set A and convex non-decreasing functions g a i (i ∈ {1, . . ., k}, a ∈ A,) such that Proposition 4.6.Assume that f satisfies (4.5).Then, for any i ∈ {1, . . ., k}, ) Proof.The proof uses in a straightforward way the computations done in the proof of Theorem 4.3.
We deduce the two following extensions of Theorem 4.3.
Proposition 4.7.Let {I a } a∈A be a partition of {1, . . ., k} and assume that where the g j 's are real-valued functions.Let X * i be a random variable independent of X and assume that X * i ≤ disp X i and X * i ≤ st X i .If log g i is a non-decreasing convex function or a non-increasing concave function, then S * Ti ≤ S Ti .Proof.Given that the I a are disjoint, this result directly follows from the proof of Theorem 4.3.Proposition 4.9.Let f (X) = ϕ 1 (X i ) j =i g j (X j ) + ϕ 2 (X i ) with log ϕ 1 and log ϕ 2 non-decreasing and convex.If Then S * Ti ≤ S Ti .The proof is postponed to Appendix B. The second condition in Proposition 4.9 is very technical and unsatisfactory.Nevertheless, very simple counter-examples exist as it can be seen below.
and, using computations made in the proof of Proposition 4.9, S * Ti > S Ti .The following result derived in [9] is mentioned here as a related result on excess wealth orders, even if it is not sufficient to obtain a more general version of Proposition 4.9.
Proposition 4.11 ([9], Cor.3.2).Let X and Y be two random variables with finite means and supports bounded from below by X and Y respectively.If X ≤ ew Y and X ≤ Y then for all non-decreasing and convex functions h 1 , h 2 for which h i (X) and h i (Y ) i = 1, 2 have second order moments, (4.9)

Examples
In this section, we illustrate the previous results on some classical financial risk models.All considered models are associated with a set of parameters.Let us first recall the conditions under which some particular distribution functions are ordered with respect to the dispersive order.We refer to [17] for other classes of distribution functions.(2) If X ∼ E(µ) and Y ∼ E(λ), then X is smaller than Y for the dispersive order (X ≤ disp Y ) if and only if λ ≤ µ.
(3) If X ∼ N(m 1 , σ 2 ) and Y ∼ N(m 2 , ν 2 ), then X is smaller than Y for the dispersive order (X ≤ disp Y ) if and only if As mentioned in Section 3, most of the numerical illustrations will be based on model parameter with truncated distribution functions.We present some properties of such distributions.Proposition 5.2.Let X and Y be two random variables (1) if X ∼ N T (m, σ 2 ) where X is truncated on [a, b] then the quantile function of X is given by and where φ is the standard normal cumulative distribution function.
where λ denotes the parameter of the exponential distribution.
The following lemma gives some conditions that ensure the ordering of two truncated random variables with respect to the dispersive order.Proof.The previous conditions can be easily derived by differentiating the difference of the quantile functions X and by using the fact that this derivative should be positive.
Also, we recall (see Rem. 4.2) that if two random variables ordered for the dispersive order have the same finite left point of their support, then they are ordered for the stochastic order.In the example that we will consider, N T (resp.E T ) denotes the truncated Gaussian (resp.exponential) distribution on [0, 2].

Value at Risk sensitivity analysis
In Section 3, we presented the Value at Risk in a simple and classical financial model.Table 1 illustrates the consistency of total Sobol indices when the distributions of input parameters are ordered with respect to the dispersive order.Indeed, each line of Table 1 corresponds to a scenario where one of the parameter has been increased with respect to both the dispersive and the stochastic order.As can be seen, changing the laws of model parameters µ and σ have a significant impact on the values of total Sobol indices S Tµ and S Tσ .Note that the ordering among Sobol indices is fully consistent with the one predicted by Theorem 4.3.

Vasicek model
In risk management, present values of financial or insurance products are computed by discounting future cash-flows.In market practice, discounting is done by using the current yield curve, which gives the offered interest rate as a function of the maturity (time to expiration) for a given type of debt contract.In the Vasicek model, the yield curve is given as an output of an instantaneous spot rate model with the following risk-neutral dynamics where a, b and σ are positive constants and where W is a standard brownian motion.Parameter σ is the volatility of the short rate process, b corresponds to the long-term mean-reversion level whereas a is the speed of convergence of the short rate process r towards level b.The price at time t of a zero coupon bond with maturity T in such a model is given by (see, e.g., [5]): where The yield-curve can be obtained as a deterministic transformation of zero-coupon bond prices at different maturities.
In what follows, we quantify the relative importance of the input parameters {a, b, σ} affecting the uncertainty in the zero-coupon bond price at time t = 0.In the following numerical experiments, the maturity T and the initial spot rate r 0 are chosen such that T = 1 and r 0 = 10%.Tables 2 and 3 report the total Sobol indices of the parameter a, b, σ under two different risk perturbations in the probability laws of these parameters.Table 2 illustrates the effect of an increase of the mean-reverting level with respect to the dispersive order and the stochastic dominance order, i.e., b * ≤ disp b and b * ≤ st b.We observe from Table 2 that the relative importance of the mean-reverting level b increases from 0.52 to 0.57.Although the total Sobol index of σ decreases, the total index of a increases.Note that the assumptions of Theorem 4.3 are not satisfied here: one can show that the output function (5.2) is log non-decreasing and log convex in b but the multiplicative form (4.4) does not hold.
Table 3 displays the total Sobol indices when σ * ≤ disp σ and σ * ≤ st σ.The law of σ is taken as a truncated Gaussian random variable on [0, 2] and has a variance of 0.3.We observe an increase in the total Sobol index of σ and a decrease in total index of a and b.

Heston model
In finance, the Heston model is a mathematical model which assumes that the stock price S t has a stochastic volatility σ t that follows a CIR process.The model is represented by the following bivariate system of stochastic differential equations (SDEs) (see, e.g., [10]) where d B, W t = ρdt.
The model parameters are • r: the risk-free rate, • q: the dividend rate, • κ > 0: the mean reversion speed of the volatility, • θ > 0: the mean reversion level of the volatility, • σ > 0: the volatility of the volatility, • σ 0 > 0: the initial level of volatility, 1]: the correlation between the two Brownian motions B and W .
The numerical computation of European option prices under this model can be done by using the fast Fourier transform approach developed in [7] which is applicable when the characteristic function of the logarithm of S t is known in a closed form.In this framework, the price at time t of a European call option with strike K and time to maturity T is given by where for j = 1, 2 Given that the input parameter are not known with certainty, which one mostly affect the uncertainty of the output pricing function (5.5).Table 4 displays the total Sobol indices of each parameter under two assumptions on the distribution of input parameter.In the first case (3 first columns), all parameters are assumed to be uniformly distributed.In the second case (3 last columns), we only change the distribution of the interest rate parameter r is such as way that r * ≤ disp r and r * ≤ st r.The option characteristics are taken as follows: T = 0.5, S 0 = 100, K = 100.As can be observed in Table 4, the influence of the input factors κ, θ, σ and ρ is negligible under the two considered assumptions.Note that most of the uncertainty in the option price is due to the dividend yield q and the interest rate r.Similar conclusions are outlined in [21] but the difference here is that we analyze how the total Sobol indices are affected by a change in the law of some input parameters.Interestingly, when the law of the interest rate parameter r * is changed to r, this parameter becomes more important than the dividend yield in terms of output uncertainty.

Multivariate extension
Here we present some multivariate extensions of the previous results.Our goal is to explore how an increase of riskness (in the sense of multivariate stochastic orders) of a group of input parameters may have an impact Table 4.Total Sobol indices for the price of a call option in the Heston model.All digits are significant with a 95% probability.

Parameter Law
Total index Parameter Law Total index on the output.We shall be mainly interested in the multivariate dispersive order and the univariate stochastic increasing convex order.We refer to [22] for a detailed review on the subject.
In the same way, for another random vector Y = (Y 1 , . . ., Y n ) with joint distribution function G, define then define recursively Definition 6.1.We say that X is smaller than Y for the multivariate dispersive order, denoted by A real function φ on R n is said to be directionally concave if, under the same condition, one has The following result proved in [22] gives conditions under which functions of two random vectors are ordered according to the st:icx order.Proposition 6.4.[22] Let X and Y be two n-dimensional nonnegative independent random vectors.If X ≤ Disp Y , then φ(X) ≤ st:icx φ(Y ) for all increasing directionally convex functions φ : R n → R As before, we consider a model output of the form Y = f (X 1 , . . ., X k ) where X 1 , . . ., X k are independent random variables.For u ⊂ {1, . . ., k}, we denote by X u the random vector (X i , i ∈ u), X −u = (X i , i / ∈ u) and |u| denotes the length of u, i.e. its number of elements.Thus, the impact of a group of variables X u , u ⊂ {1, . . ., k} on Y = f (X) may be measured by the total Sobol index (see, e.g., [12]) Using the variance decomposition formula, (6.5) can be equivalently written as Var(Y ) .(6.6)

Relationship with multivariate stochastic order: the additive case
In this subsection we suppose that f (X) can be expressed in the following additive form, i.e for all non empty set u ⊂ {1, . . ., k} f (X 1 , . . ., X k ) = g(X u ) + h(X −u ) + K (6.7) where g : R |u| → R and h : R |−u| → R are real valued functions and K ∈ R.
As in the univariate case, for u ⊂ {1, . . ., k}, X * u denotes another random vector that will be compared to X u .We suppose that for non empty u ⊂ {1, . . ., k}, the vector of input parameters X can be represented as X = (X u , X −u ).We shall assume X * u ≤ Disp X u and study the impact of replacing the group of variable X u by X * u on total Sobol indices.We assume that X * u is independent of X −u and we denote by X * = (X * u , X −u ) the vector X where X u has been replaced by X * u and by Var(Y * ) .(6.8) the total Sobol indices of X * u associated to the output Y * = f (X * ).Theorem 6.5.We assume that (6.7) is satisfied.Let X * u be a random vector independent of X −u and such that X * u ≤ Disp X u .If g is a non-decreasing directionally convex function, then S * Tu ≤ S Tu .
The proof of Theorem 6.5 makes use of Proposition 6.4.
Proof.The proof makes uses in a straightforward way of the computations done in the proof of Theorem 4.1.

Relationship with multivariate stochastic order: the product case
Here, we consider the extensions of the previous result in the case where the function f is a product of functions with several variables which fulfill some multivariate monotonicity conditions.Theorem 6.6.We assume that for a non empty set u ⊂ {1, . . ., k}, the output function f writes f (X 1 , . . ., X k ) = g(X u ) × h(X −u ) + K (6.9) where g : R |u| → R and h : R |−u| → R are real valued functions and K ∈ R. Let X * u be a random vector independent of X −u and assume that X * u ≤ Disp X u .If g is a non-increasing directionally concave function, then S * Tu ≤ S Tu .Proof.Remember that By definition since every measurable function of X u and X −u are independent (because X u and X −u are supposed to be independent random vector ).Thus using the variance decomposition formula, (6.10) can be equivalently written as

Proposition 5 . 1 .
Let X and Y be two random variables.(1)If X ∼ U[a, b] and Y ∼ U[c, d], then X is smaller than Y for the dispersive order (X ≤ disp Y ) if and only if b − a ≤ d − c.

Lemma 5 . 3 .
Let X and Y be two random variables.(1)If X ∼ U[a, b] and Y ∼ N T (m, σ 2 ) where Y is truncated on [c, d], then X is smaller than Y for the dispersive order X ≤ disp Y if and only if b − a ≤ σ √ 2π(φ(β) − φ(α))where φ represents the cumulative distribution of a standard Gaussian law and where α and β are given as in Proposition 5.2.(2) If X ∼ E T (µ) and Y ∼ E T (λ) are truncated on the same interval then X ≤ disp Y if and only if λ ≤ µ.

Table 2 .
Total Sobol indices as a result of a risk perturbation of b.All digits are significant with a 95% probability.

Table 3 .
Total Sobol indices as a result of a risk perturbation of σ.All digits are significant with a 95% probability.
3, where x i (u) and y i (u) are defined respectively in (6.2) and (6.4).Definition 6.3.In what follows, ≤ stands for the coordinatewise ordering on R n and we use the notation [x, y] ≤ z for any x, y, z ∈ R n such that x ≤ z and y ≤ z (similarly, z ≤ [x, y] if z ≤ x and z ≤ y).A real function φ on R n is said to be directionally convex if for any x